Question: The daily average temperature in Santiago, Chile, varies over time in a periodic way that can be modeled approximately by a trigonometric function. Assume the length of the year (which is the period of change) is exactly $365$ days long. The hottest day of the year, on average, is January $7$, when the average temperature is about $29^\circ C$. The average temperature at the coolest day of the year is about $14^\circ C$. Find the formula of the trigonometric function that models the daily average temperature $T$ in Santiago $t$ days into the year. ( $t=1$ on January $1$ ) Define the function using radians. $ T(t) = $ What is the average temperature in Santiago on January $31$ ? Round your answer, if necessary, to two decimal places. $ $
Let's start by writing a formula for the temperature $u$ days after January $7$. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. We know the temperature reaches its highest point at $u = 0$, so let's use a cosine function. The amplitude of the temperature function is $\dfrac{29 - 14}{2} = 7.5^\circ C$. The period is $365$ days, since a cosine function reaches its peak once in every period. The midline is the average of the highest and lowest values, or $\dfrac{29+14}{2} = 21.5$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we need to stretch it horizontally by a factor of ${\dfrac{365}{2\pi}}$, stretch it vertically by a factor of ${7.5}$, and move it up by ${21.5}$ units. $ T(u) = {7.5}\cos\left({\dfrac{2\pi}{365}}u\right) + {21.5}$ Since January $7$ is the $7$ th day of the year, the day that is $t$ days into the year is $t - 7$ days after January $7$, so $u = t - 7$ : $ T(t) = {7.5}\cos\left({\dfrac{2\pi}{365}}(t - 7)\right) + {21.5}$ Since January $31$ is $31$ st day of the year, the daily average temperature on January $31$ is about $\begin{aligned} T(31) &= {7.5}\cos\left({\dfrac{2\pi}{365}}(31 - 7)\right) + {21.5} \\ &\approx 28.37^\circ C \end{aligned}$ A correct formula for $T(t)$ is: $ T(t) = 7.5\cos\left(\dfrac{2\pi}{365}(t - 7)\right) + 21.5$ The temperature at January $31$ is: $ 28.37^\circ C$